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Set-valued discrete-time sliding-mode control of
uncertain linear systems
Felix Miranda-Villatoro, Bernard Brogliato, Fernando Castaños
To cite this version:
Felix Miranda-Villatoro, Bernard Brogliato, Fernando Castaños. Set-valued discrete-time sliding-mode
control of uncertain linear systems. IFAC 2017 - The 20th World Congress of the International
Feder-ation of Automatic Control, Jul 2017, Toulouse, France. pp.9607-9612, �10.1016/j.ifacol.2017.08.1697�.
�hal-01629185�
Set-valued sliding-mode control of uncertain linear systems:
continuous and discrete-time analysis
F´
elix A. Miranda-Villatoro
†, Bernard Brogliato
‡, and Fernando Casta˜
nos
††
Automatic Control Department, CINVESTAV-IPN, 2508 av. Instituto Polit´
ecnico
Nacional, 07360, Mexico City, Mexico.
‡
INRIA Grenoble Rhˆ
one-Alpes, University of Grenoble-Alpes, Inovalle´
e, 655 av. de
l’Europe, 38334, Saint-Ismier, France.
May 26, 2016
Abstract
In this paper we study the closed-loop dynamics of linear time-invariant systems with feedback control laws that are described by set-valued maximal monotone maps. The class of systems considered in this work is subject to both, unknown exogenous disturbances and parameter uncertainty. It is shown how the design of conventional sliding mode controllers can be achieved using maximal monotone operators (which include the set-valued signum function). Two cases are analyzed: continuous-time and discrete-time controllers. In both cases well-posedness to-gether with stability results are presented. In discrete time we show how the implicit scheme proposed for the selection of control actions makes sense resulting in the chattering effect being almost suppressed even with uncertainty in the system.
Keywords. Differential inclusions, robust control, maximal monotone maps, sliding-mode control, discrete-time systems, linear uncertain systems, Lyapunov stability.
1
Introduction
Since its appearance in the late fifties, the so-called sliding mode motion has been associated with switching control laws. The main idea arises from the behavior of the electrical relay, i.e., the input switches between a finite number of possible values depending on the region of the phase-space in which the system is evolving. This approach works well in principle, but for real-life applications some problems arise due to the intrinsic imperfections in the elements that constitute the controller, as for example: time-delays in the reaction of the components, boundaries in the operation region (finite switching frequency), etc. Among the most dangerous effects generated because of these imperfections we can find the so-called chattering effect. The consequences of the chattering effect can be catastrophic in real systems causing component degradation, poor response and in the worst case destruction of the system.
On the other hand, the closed-loop features that offer sliding mode control are very attractive: finite-time convergence, order reduction, robustness against parametric and external disturbances, simple gain tuning. For that reason many research efforts have been directed in the study of at-tenuation of the chattering effect. Among these studies we can find adaptive schemes with variable gains [42], high-order sliding modes [30], regularization techniques [46], and suitable discrete-time implementation [1, 2, 24, 25, 26, 45].
Since the work of Filippov [20] sliding-mode control systems have been associated with differential inclusions. More precisely the solutions of a dynamical system with a discontinuous right-hand side are interpreted as solutions of an associated differential inclusion. The previous work of Filippov gives
us the existence of solutions (in the sense of Filippov) for sliding mode control systems. Surprisingly, there are only a few studies that use the set-valued setting provided by Filippov for the design of the control law that will produce the sliding phenomenon [1, 2, 24, 25, 26, 45].
The objective of this paper is twofold. First, a family of set-valued controllers —which is suitable for the design of sliding mode controllers— is introduced using the so-called maximal monotone operators. The design procedure is revisited for the continuous-time context considering parametric uncertainty and external disturbances. It is shown that the set-valued approach is consistent with the classical design methodology and powerful, allowing us to face up the multivariable problem in a natural way as well as the regularization of the set-valued map. The second aim is to show step-by-step the methodology design for the discrete-time case when the set-valued maximal monotone operators are used together with the implicit scheme proposed in [1, 2, 24]. We show how this mathematical formulation is well-posed, providing a better understanding of discrete-time sliding mode systems.
This paper extends the results in [24] considering parametric uncertainty in the plant. It also shows that any maximal monotone set-valued map —different from the commonly used signum set-valued function— can be used in order to achieve the sliding regime. Moreover, the maximal monotone operators allow us to cover in one setting several well known formulations as the componentwise control or the unit vector control [44]. The mathematical framework used in this work for explaining the sliding mode phenomenon relies on differential inclusions, where (contrary to the conservative thinking of switching), we are making emphasis in the proper selection of the control values as the main tool towards chattering suppression.
This paper is organized as follows. In Section 2 we recall some preliminaries from convex analysis together with some notation. Section 3 is devoted to the design and well-posedness, in continuous-time, of set-valued controllers using maximal monotone operators. Some results concerned the robustness, against parametric and external disturbances, of the resulting closed-loop system are presented. The discrete-time counterpart is exposed in Section 4, where the use of the implicit discretization for achieving the discrete-time sliding phase is exposed, together with some stability results and the convergence of the solutions of the discrete-time closed-loop system to a solution of the continuous-time case. Finally, Section 5 depicts the effectiveness of the family of set-valued controllers proposed in Sections 3 and 4 through the use of a numerical example.
2
Preliminaries and notation
Let X be a n-dimensional linear space, dotated with the classical Euclidean inner product denoted as h·, ·i and the corresponding norm k · k. A multivalued map M : X ⇒ X is called monotone if it satisfies hy1− y2, x1− x2i ≥ 0, for all (x1, y1), (x2, y2) ∈ Graph M and it is called maximal monotone
if its graph is not contained in the graph of any other monotone map. The resolvent with index µ, (µ > 0), associated to a maximal monotone map M is a single-valued Lipschitz continuous map JMµ : X → X given as:
JMµ(x) := (I + µM)−1(x). Moreover, the resolvent JMµ is non-expansive, i.e., kJMµ(x1) − J
µ
M(x2)k ≤ kx1− x2k for all xi ∈ X,
i = 1, 2. A detailed study of the properties of the resolvent can be found in [4, 9, 38]. Related to the resolvent of M is the so-called Yosida approximation of index µ of the set-valued map M. Definition 1. The Yosida approximation of a maximal monotone map is given by
Mµ(x) = 1
µ(I − J
µ
M) (x). (1)
Roughly speaking, the Yosida approximation of M is a maximal monotone and Lipschitz continuous single-valued function which approximates the graph of M from below. Formally we have that for all x ∈ Dom M,
kMµ(x)k ≤ k Proj
and
Mµ(x) → ProjM(x)(0), as µ ↓ 0. (3)
In words, the Yosida approximation of M converges to the element of minimum norm in the closed convex set M(x). See e.g., [4, 9] for a proof of the previous statement and more properties about the Yosida approximation. The next result (taken from [4, Proposition 2, p.141]) states an important topological property concerning the graph of maximal monotone operators.
Proposition 1. The graph of a set-valued maximal monotone operator M : X ⇒ X is strongly-weakly closed in the sense that if xn → x strongly in X and if yn ∈ M(xn) converges weakly to y,
then y ∈ M(x).
Definition 2. Let f : X → R ∪ {+∞} be a proper, convex, lower semicontinuous function. The subdifferential of f at x ∈ Dom f is given by the set:
∂f (x) := {ζ ∈ X∗|hζ, η − xi ≤ f (η) − f (x), for all η ∈ X} , where X∗ refers to the dual space of X.
The proof of the following result can be found in [37].
Proposition 2. The subdifferential of a proper, convex, lower semicontinuous function is a maximal monotone operator.
Definition 3. Let f : X → R ∪ {+∞} be a proper, convex, lower semicontinuous function. The proximal map Proxf : X → X is the unique minimizer of f (w) + 12kx − wk2, that is:
f (Proxf(x)) + 1 2kx − Proxf(x)k 2= min w∈X f (w) +1 2kx − wk 2 .
Along all this work we denote the identity matrix in Rn×n as In. The set Bn := {x ∈ Rn|kxk ≤ 1}
represents the unit closed ball with center at the origin in Rn with the Euclidean norm. The
boundary of a set S is denoted bd(S). Let A ∈ Rn×m, the induced norm of the matrix A is given
by, kAkm:= supkxk=1kAxk =pλmax(A>A), where λmax(B) := maxi∈{1,...n}{λi∈ σ(B)} and σ(B)
is the spectrum of the matrix B ∈ Rn×n
. Let B ∈ Rn×nbe a symmetric matrix, B is called positive
definite, (B > 0), if for any x ∈ Rn\{0}, x>Bx > 0. It is positive semidefinite, (B ≥ 0), if x>Bx ≥ 0.
Let A = A> and B = B>be square matrices, the inequality A > B stands for A − B > 0, i.e., A − B is positive definite. Let A = A> > 0, the A-norm of a vector x ∈ Rn is given by kxk2A= x>Ax. In the case where 1 ≤ p ≤ ∞ the norm kxkp= (Pi|xi|p)
1/p
for p ∈ [1, ∞) and kxk∞:= maxi|xi|.
Proposition 3 (Schur’s complement formula). Let U = U> ∈ Rn1×n1, W = W> ∈ Rn2×n2, and R ∈ Rn1×n2 be given matrices. Then, the next three statements are equivalent,
1. U R R> W > 0. 2. U > 0, and W − R>U−1R > 0. 3. W > 0, and U − RW−1R>> 0.
3
Design of sliding mode controllers in continuous-time using
maximal monotone maps
3.1
The robust control problem
In this section we make a review of the conventional methodology design of sliding mode controllers. This review will be useful for two reasons. First, we show that the family of set-valued maximal monotone operators can be used in the design of controllers that guarantee the sliding motion.
Second, the concepts recalled here are used for introducing their discrete-time counterpart. We start analyzing a linear time-invariant system with both parametric uncertainty and external disturbances. Specifically, in this work we focus on the case in which the input matrix B ∈ Rn×m is known and
the dynamics of the plant is affected by a time and state-dependent additive uncertainty ∆A(t, x) ∈
Rn×n. The system is characterized in state-space form as: ˙
x(t) = (A + ∆A(t, x(t)))x(t) + B u(t) + w(t, x(t)), x(0) = x0, (4)
where x(t) ∈ Rn represents the state variable, u(t) ∈ Rm is the control input, w(t, x(t)) ∈ Rm ac-counts for an external disturbance considered unknown but bounded in the L∞sense. The matrix A represents the nominal values of the parameters of the plant which are assumed to be known. Notice that, in general, the addition of the term ∆A generates a state-dependent mismatched disturbance.
Along all this paper, we consider the following assumptions: Assumption 1. The pair (A, B) is stabilizable.
Assumption 2. The matrix B ∈ Rn×m has full column rank.
Assumption 3. For all t ∈ [0, +∞) the uncertainty matrix-funcion ∆A(t, ·) is locally Lipschitz
continuous and satisfies: ∆A(t, x)Λ∆>A(t, x) < In, for all x ∈ Rn and for some known symmetric
positive definite matrix Λ ∈ Rn×n.
Assumption 4. There exists W > 0 such that supt≥0kw(t, x)k ≤ W < +∞.
Notice that Assumption 3 implies that ∆A(t, x) is uniformly bounded, since k∆A(t, x)k2m≤ 1/λmin(Λ) =
λmax(Λ−1) for all (t, x) ∈ R+× Rn. It is also noteworthy that the kind of parametric disturbances
considered in this work embraces time-variant and a subfamily of nonlinear systems.
Proposition 4. Assumption 1 holds if and only if for some a > 0 there exists a symmetric positive definite matrix P ∈ Rn×n satisfying the following linear matrix inequality (LMI):
B>⊥ AP + P A>+ 2aP B⊥< 0, (5)
where B⊥∈ Rn×n−m denotes an orthogonal complement of the matrix B, i.e., B⊥ is a full column
rank matrix whose columns are formed by basis vectors of the null space of B>.
Proof. This fact follows directly from the equation (5.17) and the elimination of matrix variables procedure described in Section 2.6.2, both in [8].
The design of sliding mode controllers is made by the selection of two main elements, the sliding surface and the control law. The former refers to a submanifold on the state-space in which all the trajectories will converge in finite-time by the action of the control law, and the closed-loop system constrained to the sliding surface satisfies the performance requirements. Moreover, once the sliding surface has been reached, the task of the controller is to maintain the trajectories inside it despite the presence of disturbances (sliding phase). In this work the design of the control law is performed using a two-step design methodology. Namely, in the former stage we compute a nominal control, denoted as unom, that guarantees the invariance of the sliding surface σ = 0 in the absence of the uncertainties, i.e., w ≡ 0 and ∆A ≡ 0n×n. After that, we propose the set-valued component of
the controller, denoted by usv, which will be responsible for attaining the sliding surface as well as providing robustness against matched disturbances. A crucial point to consider is related to the proper design of the sliding surface which will guarantee the performance of the system in the sliding phase. It was proved in [14, 16, 36] that the correct design of the sliding surface helps to diminish the effects caused by mismatched disturbances and in some special cases (when some structure of the disturbance is imposed) even suppression can be accomplished [17]. More important is the fact that the wrong selection of this surface could increase the effects of the disturbance [14], which in our context implies higher gains. Throughout this work we consider the sliding surface as a hyperplane of the form σ = Cx.
Assumption 5 will guarantee the uniqueness of the equivalent control as well as the uniqueness of the nominal control. It is noteworthy that the two-step design methodology described above is sometimes called equivalent-control-based method and the part of the controller denoted by unom is
called the equivalent control. In this work the concept of equivalent control is used as in [44], i.e., it is the control that maintains the state in sliding motion in the presence of disturbances. It follows that the term unom is a nominal equivalent control, but we prefer called it merely nominal in order
to avoid confusion.
3.2
Design of the sliding surface
In this subsection we follow the lines of [14], analyzing the effect of the sliding surface σ = Cx over the mismatched disturbance. We start studying how the dynamics in sliding phase is affected by the disturbance ∆A(t, x)x. To this end we use the equivalent control method [41]. Namely, we compute
the control that maintains the sliding regime and we will see how the mismatched disturbance affects the closed-loop system. The equivalent control is computed from the invariance condition ˙σ = 0 as: C(Ax + B(ueq+ w)) + ∆A(t, xeq)xeq) = 0, ⇒ ueq= −(CB)−1C (Axeq+ ∆A(t, xeq)xeq) − w. (6)
Substitution of the equivalent control into (4) leads to the expression of the dynamics in sliding phase,
˙
xeq= In− B(CB)−1C Axeq+ In− B(CB)−1C ∆A(t, xeq)xeq, (7)
from which it becomes clear that the matrix characterizing the sliding hyperplane plays a role into the equivalent disturbance In− B(CB)−1C ∆A(t, x)x. In [14] the authors proved that the correct
design of such hyperplane guarantees the no amplification of the disturbance by using surfaces of the form C = B> or C = B+, where B+ stands for the left-inverse of the matrix B, i.e., B+ = (B>B)−1B−1. In this work we modify such selection of the surface considering instead C = B>P−1 and also C = (B>P−1B)−1B>P−1, where P is a solution of (5). First we show that this selection of C gives us an equivalent disturbance with minimum P−1-norm. Afterwards we show
how the proper choice of the matrix P dominates the mismatched disturbance in sliding phase. Lemma 1. Let C1= B>P−1 and C2= (B>P−1B)−1B>P−1, then both Ci, i = 1, 2, minimize the
P−1-norm of the equivalent disturbance (In− B(C − B)−1C)∆A(t, xeq)xeq.
Proof. Let φeq= ∆A(t, xeq)xeq. Then, the optimization problem
min C∈Rm×n In− B(CB)−1C φeq 2 P−1 = min z∈Rmkφ eq − Bzk2P−1, (8)
where z = (CB)−1Cφeq, has a unique solution given by z∗ = (B>P−1B)−1B>P−1φeq. From the definition of z it follows that C = B>P−1 achieves the minimum in (8), as well as, C =
(B>P−1B)−1B>P−1.
Notice that both selections of C stated in Lemma 1 satisfy Assumption 5. Throughout this section we will set C = (B>P−1B)−1B>P−1. In the next subsection we design the control law that assures the sliding motion.
3.3
Design of the control law
The computation of the nominal control is made from the invariance condition ˙σ = 0 in the ideal case, (i.e., w = 0, usv= 0 and ∆
A= 0), as:
˙σ = C ˙x = C (Ax + Bunom) = 0 ⇒ unom = −(CB)−1CAx. (9) Notice that the nominal control is nothing more than a linear feedback law of the form unom = −Γx with Γ = (CB)−1CA. Substitution of the nominal control (9) into the system (4) yields,
˙
where usv is the set-valued part of the controller. In order to obtain the dynamics of the system in the sliding phase, we consider the nonsingular transformation,
T = B>⊥ (B>P−1B)−1B>P−1 , T−1=P B⊥(B⊥>P B⊥)−1 B . (11)
Remark 1. It is worth to mention that from the product T−1T we obtain the identity,
P B⊥(B⊥>P B⊥)−1B⊥>+ B(B>P−1B)−1B>P−1= In. (12)
From the application of (12) to the term φ := ∆A(t, x)x it follows that
φ = P B⊥(B⊥>P B⊥)−1B>⊥φ + B(B>P−1B)−1B>P−1φ = P B⊥φu+ Bφm,
where, φu:= (B>⊥P B⊥)−1B⊥>φ and φm:= (B>P−1B)−1B>P−1φ are called the unmatched and the
matched parts of φ respectively.
The change of coordinates z = T x leads to the regular form [44], ˙ z1= B>⊥ A + ˆ∆A(t, z) P B⊥ B⊥>P B⊥ −1 z1+ B>⊥ A + ˆ∆A(t, z) Bσ (13a) ˙σ = usv+ ˆw(t, z) + ˆφm(t, z), (13b)
where, ˆ∆A(t, z) := ∆A(t, T−1z), ˆw(t, z) := w(t, T−1z) and ˆφm(t, z) := φm(t, T−1z). One comment
takes place here. First, it is easy to see that z2 = σ and from (13b) it follows that the dynamics
of the sliding variable is only affected by the matched part of the original disturbance ∆A(t, x)x.
Hence, in order to achieve the sliding regime it is necessary to take into account only the matched part of the disturbance in the design of usv [14].
In the next lines we show what are the conditions that the matrix P must satisfy such that the reduced order dynamics z1 is asymptotically stable with decay rate a, in the ideal sliding phase,
under the influence of the parametric uncertainty ∆A. To this end, let us consider the reduced order
system ˙ z1= B>⊥ A + ˆ∆A(t, z) P B⊥ B⊥>P B⊥ −1 z1, (14)
with the Lyapunov-function candidate V (z1) = 12z1>(B⊥>P B⊥)−1z1. Taking the derivative of V
along the trajectories of (14) yields, ˙ V = z1>(B⊥>P B⊥)−1z˙1 = 1 2z¯ > 1B > ⊥ AP + P A> B⊥z¯1+ ¯z1>B > ⊥∆AP B⊥z¯1, (15) where ¯z1= B⊥>P B⊥ −1
z1. Applying (5), together with the inequality 2p>X>Y q ≤ p>X>ΨXp +
q>Y>Ψ−1Y q, for some Ψ = Ψ>> 0, it follows that ˙ V ≤ −a¯z1>P ¯z1+ 1 2¯z > 1B>⊥∆AΨ∆>AB⊥z¯1+ 1 2z¯1B > ⊥P Ψ−1P B⊥z¯1. (16)
Taking Ψ = Λ where Λ = Λ>> 0 is defined in Assumption 3 gives, ˙ V ≤ −a¯z1>P ¯z1+ 1 2z¯ > 1B⊥>B⊥z¯1+ 1 2z¯1B > ⊥P Λ−1P B⊥z¯1 = −¯z1>B⊥> aP −1 2In− 1 2P Λ −1PB ⊥z¯1. (17)
From (17) asymptotic stability of the reduced system (14) in sliding phase follows if
B⊥> aP −1 2In− 1 2P Λ −1PB ⊥> 0, (18)
Along all this section we will assume that the matrix P satisfies (5) and a stronger version of (18). Namely, Q :=B > ⊥ aP − In−12P Λ−1P B⊥ −12B⊥>AB −1 2B >A>B ⊥ K −12B>Λ−1B > 0, (19) where K = K> ∈ Rm×mis a positive definite matrix introduced below. Applying Schur’s
comple-ment formula, it is easy to show that (19) is equivalent to B>⊥(aP − In) B⊥ −12B > ⊥AB B>⊥P 0n−m×n −1 2B >A>B ⊥ K 0m×n B> P B⊥ 0n×m 2Λ 0n×n 0n×n−m B 0n×n 2Λ > 0. (20)
The justification for considering (19) instead of (18) comes from the proof of the Theorem 1 where the complete system (13) is analyzed. Remark that the LMI (20) is feasible for a > 0 big enough and K, Λ sufficiently big too. This last condition translates to considering small parametric uncertainties ∆A, see Assumption 3.
Proposition 5. The disturbance term ˆφm(t, z) satisfies the linear growth condition k ˆφm(t, z)k ≤
√ κkzk, where κ = λmax(P )λmax(Λ −1) λmin(B>P−1B)λmin(P ) max 1 λmin(B>⊥P B⊥) , λmax(B>P−1B) (21)
Proof. From the definition of ˆφm we have that
k ˆφm(t, z)k = k(B>P−1B)−1B>P−1∆ˆA(t, z)T−1zk
≤ k(B>P−1B)−1B>P−1/2kmkP−1/2kmk ˆ∆A(t, z)kmkT−1kmkzk.
Recalling that the induced euclidean norm coincides with the spectral norm and making use of the Assumption 3, after simple computations we obtain,
k ˆφm(t, z)k ≤
s
λmax(Λ−1)
λmin(B>P−1B)λmin(P )
kT−1kmkzk.
On the other hand, from (11) it follows that,
kT−>k2 m≤ (B> ⊥P B⊥)−1B⊥>P1/2 B>P−1/2 2 m kP1/2k2 m = λmax(P )λmax (B> ⊥P B⊥)−1 0 0 B>P−1B ,
and the result follows. This concludes the proof. 3.3.1 Set-valued controller
In this subsection we study the family of set-valued maximal monotone operators used as feedback control laws for the system (13). First, some results about the existence and (in some cases) unique-ness of solutions are presented. Subsequently, we prove how a subfamily of the family of maximal monotone controllers yields finite-time stable sliding modes. We start setting the missing term usv
in (13b) as,
−usv(t) ∈ Kσ(t) + γ(z(t))M(σ(t)), (22)
where K ∈ Rm×m
is a positive definite matrix satisfying (20), γ : Rn → R
+ is a positive function
Thus, from (22) it follows that there exists ζ ∈ M(σ) such that −usv = Kz + γ(z)ζ. Hence, the evolution of the sliding variable is dictated by the following differential inclusion,
(
˙σ(t) = −Kσ(t) − γ(z(t))ζ(t) + ˆw(t, z) + ˆφm(t, z), σ(0) = σ0
ζ(t) ∈ M(σ(t)). (23)
In the case when the function γ is constant, the differential inclusion (23) belongs to the class of differential inclusions with maximal monotone right-hand side for which numerous results have been proposed, see e.g., [4, 6, 9, 10, 12, 33, 35] and it embraces several mathematical formulations [11]. The existence and uniqueness of solutions of (23) for the case where γ is constant has been studied for several conditions imposed on the term ˆw + ˆφ, see e.g., [9, 12, 15]. For a solution of (23) we mean an absolutely continuous function σ : R+ → Rm that satisfies σ(0) = σ0∈ Dom M together
with (23) almost everywhere on [0, +∞), that is, we consider solutions of differential inclusion (23) in the sense of Caratheodory [18]. It is worth to mention that in the case where γ is a function of the state, the uniqueness of solutions of (23) is not guaranteed, this comes from the fact that, in general, the map γ(z)M (σ) is not maximal monotone. Here, we present some examples about the different choices of the set-valued map M.
Example 1. Let M be the subdifferential of f (σ) := kσk1=P n
i=1|σi|. Then, M(σ), is the vector
set-valued signum function,
[M(σ)]i= 1, if σi> 0, [−1, 1], if σi= 0, −1 if σi< 0.
In this case the control scheme agrees with the so-called componentwise sliding mode design, see e.g., [44].
Example 2. Let M be the subdifferential of f (σ) := kσk2. Then M(σ) is the set-valued vector
function, M(σ) = ( Bn, if kσk = 0, σ kσk, otherwise.
In this case the control scheme coincides with the so-called unit vector approach [34, 39].
Example 3. Let ΨS be the indicator function of the closed convex set S, i.e., ΨS(σ) = 0, if σ ∈ S
and ΨS(σ) = +∞ otherwise. Let σ(0) be inside the set S and let M be the subdifferential of the
indicator function,
M(σ) = {ζ ∈ Rm|hζ, η − σi ≤ 0, for all η ∈ S} = NS(σ).
Here NS(σ) denotes the normal cone to the set S at the point σ. Then the closed-loop system
(13b)-(22) is well-posed and by Theorem 2 below the sliding mode is reached in finite time. The study of this kind of controllers has been reported in [31, 32]. Moreover, if S = S(t) is a Lipschitz continuous set-valued mapping, then the closed-loop system (13b)-(22) represents a perturbed Moreau’s sweeping process [13, 19].
In what follows we consider the next condition on the set-valued operator M. Assumption 6. The set-valued maximal monotone map M satisfies: 0 ∈ int M(0).
Remark 2. Assumption 6 is known as a condition for dry friction in the mechanics literature. It is strongly linked to the finite-time convergence property, see Theorem 2 and Corollary 3 below. In [3, 5] the same condition was used for proving the finite-time stability of nonlinear oscillators in both, continuous and discrete-time settings.
It is worth to mention that Assumption 6 rules out linear controllers, since we ask for maps M that must be set-valued at the origin. For example, in the case when M = ∂Φ where the function Φ is
proper, convex and lower semicontinuous, Assumption 6 ask for functions Φ which are nonsmooth at the origin, so that int M(0) 6= ∅, as for example, the norm function k · kp, 1 ≤ p ≤ ∞. This
last comment reveals that the maximal monotone operators suit perfectly as a tool that unifies the different generalizations of the signum multifunction in the design of sliding mode controllers in the multivariable case.
Proposition 6. Let Assumption 6 hold. Then for any (x, y) ∈ Graph M there exists an ε > 0 such that,
hx, yi ≥ εkxk. (24)
Proof. From Assumption 6, it follows that there exists ε > 0 such that for all ρ ∈ εBm (0, ρ) ∈
Graph M. Then, from the definition of a maximal monotone map it follows that for any (x, y) ∈ Graph M and any ρ ∈ εBm, 0 ≤ hy − ρ, xi. Consequently, supρ∈εBmhρ, xi ≤ hy, xi. The conclusion
follows.
3.4
Well-posedness and stability of the closed-loop system
In this subsection we show the well-posedness of the closed-loop system (13),(22) in the case when γ is a state-dependent gain by imposing some conditions on P , in the form of LMI’s, such that the unmatched part of the disturbance is dominated, and hence assuring the asymptotic stability of the fixed-point z∗1 = 0. After that, we show how the sliding phase is reached in finite time with an appropriate selection of the gain γ. Finally some results about stability and uniqueness of solutions in the case where γ is constant, are established.
Theorem 1. Let Assumptions 1-6 hold. Then the closed-loop system (13),(22), where M : Dom M ⇒ Rm is a set-valued maximal monotone map that satisfies Dom M = Rm, has at least one solution (in Caratheodory’s sense [18]), whenever, P = P>> 0 satisfies the LMI’s (5), (20) and in addition for some ρ > 0,
εγ(z) = ρ + W +√κkz(t)k, (25) where κ is as in (21) and ε > 0 is such that εBm⊂ M(0).
Proof. The proof follows a classical approach. Namely, first we approximate the solutions of the differential inclusion (13),(22) by using differential equations. After that, the boundedness of the solutions of the differential equation for all time t ∈ [0, +∞) is proved. Finally, the application of the Arzel`a-Ascoli [29, Theorem 1.3.8] and the Banach-Alaoglu [29, Theorem 2.4.3] theorems gives us the convergence of the sequence formed from the solutions of the differential equation to one solution of the differential inclusion (13),(22), see e.g., [3]. We start with the proof as follows, consider first the differential equation
˙ zµ1 = B>⊥A + ˆ∆A(t, zµ) P B⊥ B⊥>P B⊥ −1 z1µ+ B⊥>A + ˆ∆A(t, zµ) Bσµ (26a) ˙σµ= −Kσµ+ ˆw(t, zµ) + ˆφm(t, zµ) − γ(zµ)Mµ(σµ), (26b) where zµ = [zµ>
1 σµ>]> and the map Mµ : Rm→ Rmrefers to the Yosida approximation of index
µ > 0 of the map M (see Definition 1). It is a well known fact that the Yosida approximation is a Lipschitz continuous function with constant 1/µ. Hence, it follows that there exists one solution to (26) in [0, T ) for some T > 0. Next, using a Lyapunov analysis we show that the solution of (26) exists for all time t > 0. To this end, consider the positive definite function
V (z1µ, σµ) := 1 2z µ> 1 (B > ⊥P B⊥)−1zµ1 + 1 2σ µ>σµ, (27)
of (26) leads to, ˙ V = z1µ>(B⊥>P B⊥)−1z˙1µ+ σ µ>˙σµ = z1µ>(B⊥>P B⊥)−1B>⊥ A + ˆ∆A(t, zµ) P B⊥(B⊥>P B⊥)−1z1µ + zµ>1 (B>⊥P B⊥)−1B⊥> A + ˆ∆A(t, zµ) Bσµ− σµ>Kσµ + σµ>−γ(zµ)Mµ(σµ) + ˆw(t, zµ) + ˆφ m(t, zµ) ≤1 2z¯ µ> 1 B > ⊥(AP + P A>)B⊥z¯µ1 + ¯z µ> 1 B > ⊥ABσµ+ ¯z µ> 1 B > ⊥∆ˆA(t, zµ)P B⊥z¯1µ + ¯zµ>1 B⊥>∆ˆA(t, zµ)Bσµ− σµ>Kσµ+ σµ> −γ(zµ)Mµ(σµ) + ˆw(t, zµ) + ˆφ m(t, zµ) , (28) where, ¯zµ1 = (B⊥>P B⊥)−1z1µ. The next step consists in finding bounds for the terms that involve
the unknown matrix ˆ∆A. Using the inequality 2p>X>Y q ≤ p>X>ΨXp + q>Y>Ψ−1Y q, where
Ψ = Ψ>> 0, gives us the bounds ¯ z1µ>B⊥>∆ˆAP B⊥z¯1µ≤ 1 2z¯ µ> 1 B > ⊥∆ˆAΨ ˆ∆>AB⊥z¯1µ+ 1 2z¯ µ> 1 B > ⊥P Ψ−1P B⊥z¯1µ (29) ¯ z1µ>B⊥>∆ˆABσµ≤ 1 2z¯ µ> 1 B > ⊥∆ˆAΨ ˆ∆>AB⊥z¯1µ+ 1 2σ µ>B>Ψ−1Bσµ. (30)
Taking Ψ = Λ where Λ = Λ> > 0 satisfies Assumption 3, the substitution of (29)-(30) into (28) yields: ˙ V ≤ −¯zµ>1 B⊥> aP − In− 1 2P Λ −1P B⊥z¯1µ+ ¯z1>B⊥>ABσ − σ µ> K −1 2B >Λ−1B σµ + σµ>−γ(zµ)Mµ(σµ) + ˆw(t, zµ) + ˆφ m(t, zµ) (31) ≤ −λmin( ˜Q)kzk2− γ(zµ)σµ>Mµ(σµ) + W + √ κkzµk kσµk, (32) where ˜Q ∈ Rn×n is given as ˜ Q := B> ⊥P B⊥ −1 0 0 Im Q B> ⊥P B⊥ −1 0 0 Im > 0 (33)
and Q is defined in (19). We proceed to analyze the term hσµ, Mµ(σµ)i as follows. From the
definition of the Yosida approximation (Definition 1 in Section 2) we have that σµ = µMµ(σµ) +
JMµ(σµ), and (Jµ M(σ
µ), Mµ(σµ)) ∈ Graph M. Hence, making use of both previous facts together
with (24) in Proposition 6 yields,
hσµ, Mµ(σµ)i = µkM(σµ)k2+ hJµ M(σ µ), Mµ(σµ)i ≥ µkM(σµ)k2+ εkJµ M(σ µ)k = µkM(σµ)k2+ εkσµ− µMµ(σµ)k ≥ εkσµk + µkMµ(σµ)k (kMµ(σµ)k − ε) . (34)
Substitution of (34) into (32) results in ˙ V ≤ −λmin( ˜Q)kzµk2+ kσµk(W + √ κkzµk) − γ(zµ) εkσµk + µkMµ(σµ)k (kMµ(σµ)k − ε) ≤ −λmin( ˜Q)kzµk2− εγ(zµ) − W − √ κkzµk kσµk − γ(zµ)µkMµ(σµ)k (kMµ(σµ)k − ε). (35)
Now we continue with the proof showing that for all σµ ∈ µεB/
m the term kMµ(σµ)k − ε is
non-negative. To this end, first notice that for any v ∈ µεBm ⊂ µM(0), the resolvent JMµ at the point
v is zero. Indeed, let ε > 0 be such that εBm ⊂ M(0). Then, it follows that for any v ∈ µεBm,
resolvent it follows that kJMµ(σµ)k ≤ kσµ−vk, for all v ∈ µεBm. So, from the definition of the Yosida
approximation, taking v = µεkσσµµk, and recalling that we are analyzing the case where kσµk ≥ µε, we have, kMµ(σµ)k = 1 µkσ µ− Jµ M(σ µ)k ≥ 1 µ(kσ µk − kJµ M(σ µ)k) ≥ 1 µ kσµk − σµ− µε σ µ kσµk = 1 µ kσµk − 1 − µε kσµk kσµk = ε.
Previous developments show us that it is sufficient to consider only the case when the sliding variable σµ
∈ εµBm, (since for the case σµ ∈ εµB/ m we have already shown that (35) is strictly negative).
Hence, letting kσµk ≤ µε, and recalling that in this case Jµ M(σ
µ) = 0, it follows that Mµ(σµ) = 1 µσ
µ,
and (35) transforms into ˙ V ≤ −λmin( ˜Q)kzµk2− εγ(zµ) − W − √ κkzµk kσµk − γ(zµ)kσµk kσµk µ − ε ≤ −λmin( ˜Q)kzµk2− εγ(zµ) − W − √ κkzµk kσµk − γ(zµ)kσ µk2 µ + γ(z µ)ε2µ.
Let Lc = {zµ ∈ Rn|V (zµ) ≤ c, } be the level sets of the function V and let c > 0, be such that the
initial condition z0 ∈ Lc and rBn ⊂ Lc for some r > 0. Then γ(·) is uniformly bounded in Lc by
some ¯γ > 0, and for any z ∈ Lc\ rBn we have that
˙ V ≤ − λmin( ˜Q) − ¯ γε2µ r2 kzµk2− εγ(zµ) − W −√κkzµk kσµk − γ(zµ)kσ µk2 µ . (36) From (36) we can conclude that for all µ > 0 small enough such that
µ < r
2λ min( ˜Q)
ε2γ¯ =: µ
∗, (37)
the set Lcis positively invariant, (since ˙V < 0 in bd(Lc)), and boundedness of the trajectories in the
time interval [0, T ] follows. A classical argument by contradiction gives us the existence of solutions of (26) for all T > 0. It remains to show that for any zµ(0) = z(0) = z
0 ∈ Rn, the sequences
{zµ}
µ>0 formed by the solutions of (26) converge to a solution of (13),(22) as µ ↓ 0. Continuing
with the proof, let z0µ ∈ Rn be fixed, then there exists a c > 0 such that zµ(0) ∈ L
c, and we have
that any solution of (26) satisfies zµ∈ C([0, T ]; Rn
) for any T > 0, where C([0, T ]; Rn) refers to the Banach space of continuous functions from [0, T ] to Rn with norm kyk = supt∈[0,T ]ky(t)k. Further,
the sequence of trajectories {zµ}µ>0is uniformly bounded for all 0 < µ < µ∗where µ∗ satisfies (37)
(since the set Lc is positively invariant). On the other hand, from the assumption that the domain
of M is all Rm it follows that Mµ(σµ(t)) is uniformly bounded. Actually, from the fact that the
set Lc is a compact subset of Rn, it follows that there exist a compact subset ˜Lc ⊂ Rm, such that
σµ(t) ∈ ˜L
cfor all t ≥ 0 and all 0 < µ < µ∗, and a finite collection of open sets {Oi} ⊂ Rmsuch that:
1. ˜Lc⊂ ∪ri=1Oi,
2. For each i ∈ {1, . . . , r}, M(Oi) ⊂ biBm, for some 0 < bi< +∞.
Consequently, M(σµ(t)) ⊂ ∪r
i=1M(Oi) ⊂ maxi∈{1,...,r}biBm. Hence, invoking (2) it follows that
kMµ(σµ(t))k ≤ k Proj
M(σµ(t))(0)k ≤ maxi∈{1,...,r}bi. Therefore, from Assumption 3, together with
(26) and the conclusion about the boundedness of its solutions it follows that for any 0 < µ < µ∗, ˙zµ∈
L∞([0, T ]; Rn) is uniformly bounded. Hence, we have that the sequence {zµ}µ>0is equicontinuous.
By a direct application of the Arzel`a-Ascoli Theorem [29, Theorem 1.3.8] we get that there exists a subsequence (still denoted by) {zµ}µ>0 such that zµ → z for some z ∈ C([0, T ]; Rn) uniformly
in [0, T ]. On the other hand, because ˙zµ ∈ L∞([0, T ]; Rn), an application of the Banach-Alaoglu
Theorem [29, Theorem 2.4.3] gives us that there exists a function q ∈ L∞([0, T ]; Rn), such that
˙
zµ→ q in the weak* topology, i.e.,
lim
µ↓0
Z T
0
h ˙zµ(t) − q(t), s(t)idt = 0, for all s ∈ L
1([0, T ]; Rn).
Moreover, from the fact that z(t) = z(0) +RT
0 q(t)dt we infer that q = ˙z almost everywhere. Notice
that since the considered time domain is bounded, we have that L2([0, T ]; Rn) ⊂ L1([0, T ]; Rn)
[28, Corollary 1, Chapter VIII]. Hence, ˙zµ converges weakly in L
2([0, T ]; Rn). From the continuity
assumption of ˆ∆A and the convergence of zµ and ˙zµ to z and ˙z respectively, it becomes clear that
z satisfies (13a). In fact,
˙ z1µ= B⊥>(A + ˆ∆A(t, zµ))P B⊥ B⊥>P B⊥ −1 z1µ+ B>⊥(A + ∆A(t, zµ)) Bσµ→ B⊥>(A + ˆ∆A(t, z))P B⊥ B⊥>P B⊥ −1 z + B⊥>(A + ∆A(t, z)) Bσ = ˙z1. Additionally, setting θµ:= ˙σµ+ Kσµ− ˆw − ˆφ
mwe have that for any ϕ ∈ L2([0, T ]; Rm),
Z T 0 θµ(t) γ(zµ(t))− θ(t) γ(z(t)), ϕ(t) dt = Z T 0 1 γ(zµ(t))− 1 γ(z(t)) hθµ(t), ϕ(t)i dt +Z T 0 θµ(t) − θ(t) γ(z(t)) , ϕ(t) dt
From (25) if follows that γ(z) > ρε for any z ∈ Rn. Thus, there exists a µ > 0, such that for all
µ ≤ µ∗, we have, Z T 0 θµ(t) γ(zµ(t))− θ(t) γ(z(t)), ϕ(t) dt ≤ Z T 0 ε2 ρ2Lγkz µ (t) − z(t)kkθµ(t)kkϕ(t)kdt + Z T 0 ε ρhθ µ (t) − θ(t), ϕ(t)i dt, (38) where Lγ > 0 refers to the Lipschitz constant of the function γ. Hence:
ζµ:= ˙σ µ+ Kσµ− ˆw(t, zµ) − ˆφ m(t, zµ) γ(zµ) → ˙σ + Kσ − ˆw(t, z) − ˆφm(t, z) γ(z) =: ζ, as µ ↓ 0, (39) weakly in L2([0, T ]; Rm) for any T > 0. Finally, from [4, p. 146] it follows that the set-valued map
M seen as a set-valued map from L2([0, T ], Rm) to the subsets of L2([0, T ], Rm) is also maximal
monotone. Thus, since JMµ(σµ
) → σ uniformly in C([0, T ], Rm), [4, p.144] (and consequently strongly
in L2([0, T ]; Rm)), the left-hand side of (39) is equal to ζµ = Mµ(σµ) and Mµ(σµ) ∈ M(J µ M(σ
µ)),
[4, p. 144]. Invoking Proposition 1 in Section 2 allows us to conclude that ζ ∈ M(σ), that is, the differential inclusion (13),(22) is satisfied. This finishes the proof.
Remark 3. Notice that the assumption Dom M = Rmrules out multivalued controllers with compact
domain as those introduced in Example 3. However, the use of set-valued maps whose domain is not all Rm is possible using γ > 0 constant, since we fall in the case of differential inclusion with maximal monotone right-hand side, see e.g., [9, 15].
Theorem 2. Let the assumptions of Theorem 1 hold. Then, the subsystem (13b) with set-valued controller (22) is globally finite-time Lyapunov stable whenever,
εγ(z) ≥ ρ + W +√κkzk (40) where ε is given in (24), and ρ > 0 is an arbitrary constant.
Proof. We consider the positive definite function of σ, V (σ) = 12σ>σ. From the proof of Theorem 1 we have that z1 is bounded. So, differentiating V along the trajectories of (13b) results in ˙V =
σ>˙σ = σ>(usv+ w + φ
m). From (22) there exists a ζ ∈ M(σ) such that usv = −Kσ − γ(x)ζ and
then,
˙
V ≤ −σ>Kσ − γ(z)σ>ζ + kw + φmkkσk
≤ − εγ(z) − W −√κkzk kσk,
where we have used (24) and the fact that K > 0. Hence, if (40) holds, then ˙V < −ρkσk. Finally, after integration of both sides of the last inequality an upper-bound for the time t∗such that σ(t) = 0 for all t ≥ t∗, is obtained as: t∗≤p2V (0)/ρ.
It is worth to mention that Theorem 2 does not make mention to the uniqueness of solutions, but we have proved instead that all the solutions converge to the sliding surface. The next step consists in showing the asymptotic stability of the whole system (13),(22).
Theorem 3. Let the assumptions of Theorem 1 hold. Then, the closed-loop system (13), (22) is globally asymptotically stable.
Proof. Consider the Lyapunov-function candidate, V (z1, σ) := 1 2z > 1(B > ⊥P B⊥)−1z1+ 1 2σ >σ. (41)
Let ζ be an element in M(σ), differentiating (41) along the system’s trajectories yields ˙ V ≤ −λmin( ˜Q)kzk2+ σ> −γ(z)ζ + ˆw(t, z) + ˆφm(t, z) ≤ −λmin( ˜Q)kzk2− εγ(z) − (W + √ κkzk) kσk < 0, (42) where the matrix ˜Q = ˜Q> > 0 is defined in (33) and we made use of (24). This concludes the proof.
An important case arises when we ask for a constant gain γ > 0. In this case the existence of solutions has been deeply studied (see, e.g., [9], [15], [19]) and from the practical point of view, we sacrifice the global stability for semi-global stability and the uniqueness of solutions is retrieved. Corollary 1. Let the Assumptions 1-6 hold, let α > 0, δ > 0 and P = P> be such that (5), (20) hold, and let Lc⊂ Rn be a compact set specified below in the proof. Then, for each initial condition
that satisfies (z1(0), σ(0)) ∈ Lc, for some c > 0, the closed-loop system (13) with set-valued controller
−usv∈ Kσ + γM(σ), (43)
where K = K>> 0 satisfies (19), is semi-globally asymptotically stable whenever εγ ≥ β + W +√κ max
z∈Lc
{kzk} , (44)
where z = [z>1, σ>]>, κ is given in (21), and β > 0 is an arbitrary constant.
Proof. Consider the positive definite function V (z1, σ) as in (41) and let Lc:= {(z1, σ) ∈ Rn|V (z1, σ) ≤
c} be the level sets of the function V . As first step we prove the positive invariance of the set Lc. To
this end we take the time derivative of V along the system trajectories, yielding again (42) changing γ(z) by γ. Hence, in the light of (44), we can conclude that ˙V < 0 for all σ ∈ bd(Lc) and the
positive invariance follows. Now, let (z1(0), σ(0)) ∈ Lc for some c > 0, then from (42) and the fact
that the maximum in (44) is attained in the boundary of Lc it follows that ˙V < 0 and we arrive at
From Corollary 1 it follows that the multivalued controller (43) carries the system (13) into the sliding surface σ = 0 in finite time. Moreover, as a consequence of the maximal monotonicity of the set-valued map γM(·) we have uniqueness of solutions of the closed-loop system (13), (43). Indeed, consider the following differential inclusion
˙ z ∈ f (t, z) − γN(z), (45) where: f (t, z) = " B⊥>A + ˆ∆A(t, z) P B⊥ B⊥>P B⊥ −1 B>⊥A + ˆ∆A(t, z) B 0 −K # z1 σ + 0 ˆ w(t, z) + ˆφm(t, z)
is a locally Lipschitz function in its second argument and N : Rn ⇒ Rn is a maximal monotone set-valued map described by z 7→ [0, ζ>]> and ζ ∈ M(σ). Then, assuming that there exist two solutions of (45) denoted by z1 and z2, it follows that,
d dt 1 2kz 1− z2k2= h ˙z1− ˙z2, z1− z2i = hf (t, z1) − f (t, z2), z1− z2i − γhη1− η2, z1− z2i ≤ Lfkz1− z2k2, where ηi∈ N(zi), i = 1, 2 and L
f refers to the Lipschitz constant of the function f . The application
of Gronwall’s inequality leads to
kz1(t) − z2(t)k ≤ kz1(0) − z2(0)keLft, for all t ≥ 0, making evident the uniqueness of solutions.
It is a well known fact that in the continuous-time setting the selection of the values that maintain the sliding regime depends explicitly on the values of the disturbances ˆw and ˆφmwhich are by definition
unknown. For that reason, in practical applications it is common to use a regularized version of the controller (22), leading us to the concept of boundary layer control [42]. In general, the regularization is made in an arbitrary way. In our context the regularization is well defined through the use of the Yosida regularization and as was shown in the proof of Theorem 1 this approach leads to trajectories that are in a neighbourhood of one solution of the differential inclusion (13). In the sequel we present an example for the case of the unitary vector approach.
Consider the set-valued map M as in the Example 2 and a constant gain γ > 0. From the proof of Theorem 1, it follows that our regularized control is given by the maximal monotone single valued map Mµ, which in this case is given by
Mµ(σ) = ∇fµ(σ) = 1 µ(σ − Proxµf(σ)) = ( σ kσk, if kσk > µ, 1 µσ, otherwise. (46)
It is worth to mention that (46) differs from the commonly used regularization kσk+ρσ , with ρ > 0 sufficiently small. Therefore, in the maximal monotone approach we have a unique way of computing the regularized controller coming from a set-valued maximal monotone map leading to a closed-loop system whose trajectories converge into a neighborhood of the origin. In the next section we shall study the design of this kind of maximal monotone controllers in the discrete-time setting.
4
Design of discrete-time sliding-mode controllers using
max-imal monotone maps
In this section we present a methodology for the digital implementation of discrete-time sliding mode controllers using maximal monotone maps. The design process is revisited step-by-step in order to
show how the implicit discrete-time scheme proposed in [1, 2] allows us to make a proper selection of the values of the control input at each sampling instant, and consequently reduces drastically the chattering effect at higher sampling rates.
4.1
The plant representation
We start considering the discrete-time model of (4) through the use of the Euler’s method, i.e., we take a constant sampling time tk+1− tk= h > 0 for all k ≥ 0, and we obtain,
xk+1= (In+ hA)xk+ hB(uk+ wk) + h∆Axk. (47)
It is worth to mention that in the absence of the parametric disturbances (∆A(t, x) ≡ 0), the system
(47) becomes linear and the ZOH (Zero-Order Hold) method can be applied in order to obtain the equations of the dynamics in discrete time. Using the ZOH method has the disadvantage that for big sampling times, the resulting discrete-time system could result in an uncontrollable system [27]. This disadvantage is not present when the Euler’s method is applied. Namely, assume that for a linear (unperturbed) continuous-time system the pair (A, B) is controllable, i.e., rank([λIn+ A|B]) = n for
all λ ∈ σ(A). Then, after applying the Euler’s method, the system matrices become (In+ hA, hB).
The condition for controllability of this new pair translates into, rank[µIn− (In+ hA)|hB] = n for
all µ ∈ σ(In+ hA), which is trivially satisfied in the light of µi= 1 + haλi, for all i = 1, . . . , n and
the assumed controllability of the original continuous-time system. Previous lines shows that using the Euler’s discretization method leads to controllable systems but unfortunately we lose the exact representation of the discrete-time dynamics and also it is not possible to obtain an arbitrary desired decaying rate a (see Proposition 4) which in our context translates into considering high sampling rates for the domination of the unmatched disturbance. Along all this section we also consider that Assumptions 1 through 6 hold. In the discrete-time context the counterpart of Proposition 4 is given as:
Proposition 7. Assumption 1 implies that for some a > 0 such that 0 < 2ha < 1, there exists a symmetric positive definite matrix X ∈ Rn×n satisfying the matrix inequality:
B>⊥ AX + XA>+ 2aX B⊥+ hB⊥>
XA>B⊥ B⊥>XB⊥ −1
B>⊥AXB⊥< 0. (48)
Proof. Stabilizability of the system (47) is equivalent to the existence of a matrix K ∈ Rm×n such
that for any 2ha ∈ (0, 1), there exists a matrix, U ∈ Rn×n, U = U>> 0 satisfying the discrete-time
Lyapunov’s equation:
(1 − 2ha)U − (I + hA − hBK)>U (I + hA − hBK) > 0. Pre and post multiplying by U−1 and setting W = KU−1 yields,
−h(2aU−1+ AU−1+ U−1A>− BW − W>B>) − h2 AU−1− BW>
U AU−1− BW > 0. Hence, applying the Schur’s complement formula we obtain the LMI
−h(2aU−1+ AU−1+ U−1A>− BW − W>B>) h(U−1A>− W>B)
h(AU−1− BW ) U−1
> 0.
Recalling that B⊥∈ Rn×(n−m) has full column rank, it follows that the previous inequality implies
−hB>
⊥(2aU−1+ AU−1+ U−1A>)B⊥ hB>⊥U−1A>B⊥
hB⊥>AU−1B⊥ B⊥>U−1B⊥
> 0, (49) where we have applied the full row rank congruence transformation
B⊥> 0n−m×n
0n×n−m B⊥>
∈ R2(n−m)×2n.
Finally, applying once again the Schur’s complement formula to (49) and setting X = U−1we obtain the desired result.
From (48) it is easy to see that as h ↓ 0 the solution of the matrix inequality (48) approaches the solution of the LMI (5).
To finish this subsection we compute a bound for ∆Athat will be useful in the forthcoming sections.
Proposition 8. Let X = X>> 0 be such that
X − In > 0, (50)
then,
Λ−1− ∆>AB⊥(B⊥>XB⊥)−1B⊥>∆A> 0. (51)
Proof. From Assumption 3 together with the bound on X imposed by (50) it follows that ∆AΛ∆>A<
X. Since B⊥has full column rank, it follows that B⊥>XB⊥− B⊥>∆AΛ∆>AB⊥> 0. Using the Schur’s
complement formula we obtain,
B>
⊥XB⊥ B⊥>∆A
∆>AB⊥ Λ−1
> 0,
and applying once again the Schur’s complement formula we obtain the desired result.
In the sequel we will assume that X satisfies (48) together with (50) and consequently (51) also holds.
4.2
Design of the sliding surface
In this subsection the methodology for the design of the sliding surface mimics its continuous coun-terpart. First, we start with a sliding manifold of the form σk = Sxk and conditions on the matrix
S are derived. In fact, it is shown that the resulting hyperplane has the same structure as its continuous-time analog C. We make the following assumption,
Assumption 7. The product SB is nonsingular.
Analogous to the continuous-time context, we start computing the equivalent control in order to see how the disturbance affects the sliding regime. In the discrete-time case, the necessary sliding condition ˙σ = 0 is transformed into the fixed-point condition σk+1= σk from which we obtain the
equivalent control as1:
ueqk = 1 h(SB)
−1(σ
k− S(In+ hA)xk− hS∆Axk) − wk (52)
Notice that the fixed-point condition σk+1= σk is usually neglected and changed for the condition
σk+1= 0. We will see that the fixed-point condition is well fitted for the estimation of the control
law that will achieve the sliding motion. The equivalent closed-loop dynamics in sliding mode results in: xeqk+1= In− B(SB)−1S (In+ hA)x eq k + B(SB) −1σ k+ h In− B(SB)−1S ∆Axk. (53)
From (53) it becomes clear that the structure of the sliding surface will be the same as in the continuous-time framework, i.e., throughout this section we set S = (B>X−1B)−1B>X−1 . Notice that the both surfaces (C and S) are not exactly the same since P satisfies (5) and X satisfies (48) instead, but S tends to C as h decreases to zero.
1As alluded to above, what we call the equivalent control here, is not the same as what is called the equivalent
4.3
Controller design
In this subsection we follow the discrete version of the two-steps design methodology used in the previous section. The main difference with the continuous part relies on the discretization scheme used for the control usv. It is shown that the implicit discretization approach inherits the robustness
provided by the maximal monotone operators presented in Section 3. The first step consists in computing the nominal control using the fixed-point condition σk+1= σk leading to
unomk = 1 h(SB)
−1(σ
k− S(In+ hA)xk) . (54)
Substitution of (54) into the discrete-time dynamics (47) yields
xk+1= In− B(SB)−1S (In+ hA)xk+ B(SB)−1σk+ hB(usvk + wk) + h∆Axk. (55)
Consider the coordinates transformation zk = T xk with T given in (11) but changing the matrix P
by its discrete-time counterpart X. Hence, after simple computations we get the closed-loop system in regular form, ( zk+11 = B⊥>(In+ hA + h∆A)XB⊥ B⊥>XB⊥ −1 zk1+ B⊥>(In+ hA + h∆A)Bσk (56a) σk+1= σk+ h(usvk + wk+ ηkm), (56b)
where the term ηm
k refers to the matched part of the disturbance ∆Axk, i.e., ηmk = S∆AT−1zk =
(B>X−1B)−1B>X−1∆AT−1zk, see Remark 1. It is noteworthy that system (56) is the
discrete-time counterpart of (13). It is clear that the disturbance term ηm
k satisfies a linear growth condition
similar to that associated with the term φm. Thus the following holds.
Proposition 9. The disturbance term ηm
k satisfies the linear growth condition kη m k k ≤ √ ¯ κkzkk, where ¯ κ := λmax(X)λmax(Λ −1) λmin(B>X−1B)λmin(X) max 1 λmin(B⊥>XB⊥) , λmax(B>X−1B) . (57)
4.3.1 The set-valued controller
We continue with the design of the multivalued part of the controller. The main difference with the continuous-time part relies here, where, because of the discretization method employed, it is possible to make a selection for the values of the controller that will compensate for the disturbances that affect the resulting closed-loop system. Specifically, we use the implicit Euler’s method and we show how the system automatically makes the selection of the values that will compensate for the disturbance. As a motivation of the implicit scheme used, we study first the following equivalent controller,
−usvk ∈ γM(σk+1), (58)
where γ > 0 is considered constant. Two important questions arise: is the proposed set-valued controller (58) non-anticipative? and why is it called equivalent? The label equivalent is because, in the sliding phase, usv
k is equal to u eq k − u
nom
k , i.e., the control action uk = unomk + u sv
k , with u sv k
satisfying (58), coincides with the equivalent control (52). Indeed, consider the closed-loop system (56b), (58). It follows that,
σk− σk+1+ h(wk+ ηk) ∈ hγM(σk+1) ⇔ σk+ h(wk+ ηk) ∈ (I + hγM)(σk+1)
⇔ σk+1= (I + hγM)−1(σk+ h(wk+ ηk))
⇔ σk+1= JγMh (σk+ h(wk+ ηk)), (59)
where Jh
γMrefers to the resolvent of the maximal monotone map γM of index h. Hence, the
discrete-time closed-loop dynamics of the sliding variable results in the difference equation (59). An explicit expression for the controller is obtained after substitution of (59) into (56b) as
usvk = −1 h(I − J h γM)(σk+ h(wk+ ηmk)) = −M h γ(σk+ h(wk+ ηmk )) . (60)
where the map Mhγ refers to the Yosida approximation of the set-valued map γM of index h. At
this point it is worth to mention that the selection process was done automatically by the system, i.e., the closed-loop system selects one and only one input from the maximal monotone map M in order to compensate for the disturbance term wk+ ηmk. Thus, in ideal sliding mode σk+1= σk = 0
implies usv k = −
1 h(I − J
h
γM)(h(wk+ ηkm)). Now, assuming that wk+ ηkm ∈ γM(0) it follows that
usvk = −wk− ηkm, (since J h
γM(w) = 0 for all w ∈ γM(0)). Therefore, uk = unomk + u sv k = u
eq k . The
previous development reveals that the implicit controller (58) makes sense.
Now we introduce the missing term usvk using an implicit approach, which has been studied theoret-ically in [1, 2, 24] and tested experimentally in [25, 26, 45] showing to be a very efficient way to deal with the chattering effect. It is clear that in a real implementation setting the selection procedure cannot be achieved directly, because if we try to mimic the same steps presented in the previous situation, we will have to impose the unreal assumption that we know perfectly the disturbance term wk+ ηkm, see (60). Therefore some modification to the discrete-time controller (58) must be done.
Roughly speaking, we consider the discrete-time scheme proposed in [1, 2, 24] by creating a virtual nominal system from where the selection process is achieved. Next, the controller computed from the virtual nominal system is applied to the original discrete-time plant. Formally, instead of (56), (58), we consider the extended system,
zk+11 = B⊥>(In+ hA + h∆A)XB⊥ B⊥>XB⊥ −1 zk1+ B⊥>(In+ hA + h∆A)Bσk (61a) σk+1= ˜σk+1+ h(wk+ ηmk ) (61b) ˜ σk+1= σk+ husvk (61c) −usv k ∈ K ˜σk+1+ γM(˜σk+1), (61d)
where K ∈ Rm×m is a symmetric positive definite matrix specified below. System (61) represents
the implementable discrete-time dynamics associated with the real continuous-time system (13). The variable ˜σk+1may be seen as the state of a nominal, undisturbed system, or as a dumb variable
allowing to calculate the controller usv
k. In this approach the control selection is made using the
virtual undisturbed system (61c)-(61d), and the perturbation term is implicitly taken into account through the use of the real state σk in (61c). Following the same steps as in (59), we have
σk− ˜σk+1∈ hK ˜σk+1+ hγM(˜σk+1) ⇔ σk∈ (I + h(K + γM)) (˜σk+1)
⇔ σ˜k+1= (I + h(K + γM)) −1
(σk)
⇔ σ˜k+1= JNh(σk), (62)
where K = K> > 0 is an m × m matrix and the set-valued map N := K + γM that maps p 7→ {q ∈ Rm|q = Kp + γζ, ζ ∈ M(p)} is also maximal monotone [38, Exercise 12.4]. It follows from (61c) that the input selection applied to the system is explicitly given by
usvk = −
1 h I − J
h
N (σk) =: −Nh(σk), (63)
where Nh refers to the Yosida approximation of N of index h. Equation (63) shows the
non-anticipation and the uniqueness of the control (61d) (since Nhis single valued). Hence, the
discrete-time closed-loop subsystem (61b)-(61d) is equivalent to, (
σk+1= ˜σk+1+ h(wk+ ηmk ),
˜
σk+1= JNh(σk).
(64)
In this context the variable ˜σk is called the discrete sliding variable and when ˜σk+n = 0 for all n ≥ 1
and some k < +∞, we say that the system is in the discrete-time sliding phase [24].
4.4
Stability of the closed-loop
In this section the stability of the system (61) is proved. We start by computing the necessary con-ditions that the matrices X and K must satisfy under the assumption of ideal sliding phase, that is,
σk= 0. This step allows us to compare the discrete-time and the continuous-time approaches
show-ing their similarities, and also providshow-ing some convergence results. To this end, we start considershow-ing the following discrete-time reduced order system:
z1k+1= B>⊥(In+ hA + h∆A)XB⊥ B⊥>XB⊥ −1
z1k, (65) together with the Lyapunov-function candidate V (zk1) = 12z1>k B⊥>XB⊥
−1
zk1. Computing, the difference ∆V := V (zk+11 ) − V (zk1) along the trajectories of (65) and setting G := B⊥>XB⊥ and
sk := G−1zk1, yields ∆V = 1 2z 1> k+1 B > ⊥XB⊥ −1 z1k+1−1 2z 1> k B > ⊥XB⊥ −1 zk1 =1 2s > kB > ⊥X (In+ hA + h∆A)>B⊥G−1B⊥>(In+ hA + h∆A) XB⊥sk− 1 2s > kGsk =h 2s > kB⊥> AX + XA>+ hXA>B⊥G−1B⊥>AX B⊥sk+ hs>kB⊥>∆AXB⊥sk + h2s>kB⊥>XA>B⊥G−1B>⊥∆AXB⊥sk+ h2 2 s > kB>⊥X∆>AB⊥G−1B⊥>∆AXB⊥sk. (66)
Making use of the inequality 2p>W>Y q ≤ p>W>ΨW p + q>Y>Ψ−1Y q, where Ψ = Ψ> > 0, gives us the bounds s>kB⊥>∆AXB⊥sk≤ 1 2s > kB>⊥∆AΨ1∆>AB⊥sk+ 1 2s > kB⊥>XΨ−11 XB⊥sk (67) s>kB>⊥XA>B⊥G−1B⊥>∆AXB⊥sk≤ 1 2s > kB>⊥XA>B⊥G−1Ψ2G−1B⊥>AXB⊥sk +1 2s > kB⊥>X∆>AB⊥Ψ−12 B > ⊥∆AXB⊥sk. (68)
Setting Ψ1 = Λ where Λ is any positive definite matrix that satisfies Assumption 3, and Ψ2 = G,
then applying the results from Propositions 7 and 8 transforms (66) into
∆V ≤ −hs>kB>⊥ aX −1 2In− 1 2XΛ −1X − hXΛ−1X − h 2XA >B ⊥ B⊥>XB⊥ −1 B>⊥AX B⊥sk. (69) Therefore, ∆V < 0 if and only if
B⊥> aX − 1 2In− 1 2XΛ −1X − hXΛ−1X − h 2XA >B ⊥ B⊥>XB⊥ −1 B⊥>AX B⊥ > 0. (70)
Notice the resemblance of (70) with (18). In fact, it is easy to see once again that X → P as h ↓ 0 where P is a solution of (18). Similarly to the continuous-time case, we will ask for a stronger version of (70). Namely, ¯ Q := ¯ Q11 −12B>⊥AB − h2B > ⊥XA>B⊥G−1B⊥>AB −1 2B >A>B ⊥−h2B >A>B ⊥G−1B⊥>AXB⊥ Q¯22 > 0, (71) where ¯Q11 := B⊥> aX − In−12XΛ−1X − h 2XΛ−1X + XA>B⊥ B⊥>XB⊥ −1 B⊥>AXB⊥ and ¯ Q22 := K − 12B >Λ−1B − hB> 2Λ−1+3 2A >B
⊥G−1B⊥>A B. It is also worth to notice that as h
decreases to zero, a solution (X, K) of the matrix inequality (71) tends to a solution of the matrix inequality (19). Additionally, in analogy with the continuous-time context, a series of application of the Schur’s complement formula gives us the equivalence between the matrix inequality (71) and the following LMI,
R11 R12
R>12 R22
where, R11:= B⊥>(aX − In) B⊥ −12B > ⊥AB −hB⊥>XA>B⊥ −1 2B >A>B ⊥ K −hB>A>B⊥ −hB> ⊥AXB⊥ −hB>⊥AB 2hB⊥>XB⊥ R12:= −hB> ⊥XA>B⊥ 0 B⊥>X 0 0 −hB>A>B⊥ 0 B> 0 0 0 0 R22:= 2hB>⊥XB⊥ 0 0 0 0 hB> ⊥XB⊥ 0 0 0 0 2 1+2hΛ 0 0 0 0 2 1+2hΛ .
Assumption 8. Along all this section we will assume that X and K are such that (48), (50) and (72) hold.
The following result is about the conditions in the state for achieving the discrete-time sliding phase (˜σk+1= ˜σk = 0 for all k ≥ k∗ for some 0 < k∗< +∞).
Lemma 2. Let Assumption 6 hold. The following two statements are equivalent: 1) σk∈ hγM(0) for some k ∈ N.
2) ˜σk+1= 0.
In addition, if for some k0∈ N, ˜σk0+1= 0, then ˜σk0+n= 0 for all n ≥ 1, whenever wk+η
m
k ∈ γM(0)
for all k ≥ k0.
Proof. The equivalence between 1) and 2) is clear from (64). Namely, ˜σk+1 = 0 is equivalent to
Jh
N(σk) = 0 which in fact is the same as σk∈ (I + h(K + γM))(0). For the second part of the proof
we start assuming that for some k0∈ N, ˜σk0+1= 0. Hence, again from (64) it follows that, σk0+1= ˜σk0+1+ h(wk0+ η
m
k0) = h(wk0+ η
m
k0) ∈ hγM(0). (73) Therefore, applying the first part of the lemma we obtain ˜σk0+2 = 0. The results follows by induction.
The following result supports the use of the scheme proposed in [1, 2]. Corollary 2. Let the matched disturbance wk+ ηkm∈ γM(0) for all k ≥ k
∗for some 0 < k∗< +∞.
Then, in the discrete-time sliding phase the control input usv
k satisfies:
usvk = wk−1+ ηk−1m .
Proof. Since in sliding phase ˜σk+1= ˜σk = 0 it follows from (63) that usvk = − σk
h and from (64) we
have that σk = h(wk−1+ ηk−1m ) and the result follows.
In words, the input obtained from the implicit scheme (61) compensates for the disturbance with a delay of one step once the discrete-time sliding phase has been reached. Moreover, it is worth to notice that in the discrete-time sliding phase the input usv
k is independent of the gain γ, a crucial fact
that is experimentally verified in [25, 26]. This last property becomes fundamental in the application of the control scheme (61) since it helps to drastically reduce the chattering effect of the closed-loop system.
Remark 4. It is worth to mention that the scheme proposed in [1], [2] and stated in (61) for the computation of the control input seems to be connected to the approach of integral sliding modes for the estimation of the disturbance [43]. Indeed, we can see that equation (61c) represents some sort of nominal system from which the control input is obtained instead of using the perturbed system (61b). Moreover, Corollary 2 confirms that, as a consequence of taking the implicit discretization, the obtained controller is automatically compensating the matched disturbance terms with a one-step delay.
Practical stability of the difference equation (61) is proved by the following theorem.
Theorem 4. Let Assumptions 1-7 hold. Consider the closed-loop system (61) where X = X>> 0 and K = K> > 0 are such that Assumption 8 holds. In addition, let Lc ⊂ Rn be the compact set
Lc:= z1 σ ∈ Rn 1 2z 1> B> ⊥XB⊥ −1 z1+1 2σ >σ ≤ c2 . (74)
Then, for any initial condition z0=z01> σ>0
>
which lies in Lc for some c > 0, there exists h > 0
small enough and fixed such that for γ > 0 satisfying:
γε ≥ β + W + (√¯κ + 2hkKk2)¯z, (75) where ¯z := 2c2/R is an upper bound of z
k in Lc and R := min n 1 λmax(B⊥>XB⊥), 1 o , the discrete-time closed-loop system (61a)-(61d) is semi-globally practically stable. In fact, for any initial condition z0 ∈ Lc the trajectories converge to the set Lc∗ where c > c∗ :=
q h lR
2λmin( ˆQ)
with l > 0 a constant specified below in the proof.
Proof. Mimicking (41), let us consider the Lyapunov function candidate Vk(z1, σ) = Vk
z1+Vσk, where Vk z1 := 1 2z 1> k (B⊥>XB⊥)−1zk1 and Vσk := 1 2σ > kσk. Let ∆V = ∆Vz1+ ∆Vσ where ∆Vσ := Vσk+1− Vσk
and ∆Vz1 := Vzk+11 − Vzk1. We split the proof in two parts. The first part consists in finding a proper upper-bound for the difference ∆Vσ. After this, we continue analyzing the term ∆Vz1. Finally we
put all terms together and the practical stability follows. Consider the positive definite function Vk ˜ σ = 1 2σ˜ >
kσ˜k and its respective difference ∆Vσ˜= V˜σk+1− Vσ˜k. Then, making use of (61c) and (61d)
it follows that, ∆Vσ˜= 1 2σ˜ > k+1σ˜k+1− 1 2σ˜ > k˜σk = 1 2σ˜ > k+1(˜σk+1− σk) − 1 2˜σ > kσ˜k+ 1 2σ˜ > k+1σk = ˜σk+1> (˜σk+1− σk) − 1 2˜σ > kσ˜k+ ˜σk+1> σk− 1 2σ˜ > k+1σ˜k+1 ≤ −h˜σ>k+1(K ˜σk+1+ γζk+1) + Vσk− V k ˜ σ, (76)
where ζk+1 ∈ M(˜σk+1) and we have used the inequality 2˜σ>k+1σk ≤ ˜σ>k+1σ˜k+1+ σk>σk in the last
step. Adding and subtracting the term Vk+1 σ + V k+1 ˜ σ in (76) yields ∆Vσ˜≤ −h˜σ>k+1K ˜σk+1− hγ ˜σ>k+1ζk+1+ 1 2σ > k+1σk+1− 1 2σ˜ > k+1σ˜k+1+ ∆V˜σ− ∆Vσ,
which, after substitution of (61c) into (61b) leads to, ∆Vσ≤ −h˜σk+1> K ˜σk+1− hγ ˜σ>k+1ζk+1− 1 2σ˜ > k+1σ˜k+1 +1 2(˜σk+1+ h (wk+ η m k )) > (˜σk+1+ h (wk+ ηmk )) = −h˜σk+1> K ˜σk+1− hγ ˜σ>k+1ζk+1+ h˜σk+1> (wk+ ηmk ) + h 2kw k+ ηkmk 2. (77)
From (61c) and (61d) it follows that ˜σk+1= σk− hK ˜σk+1− hγζk+1, with ζk+1∈ M(˜σk+1). Then
(77) transforms into, ∆Vσ≤ −h (σk− hK ˜σk+1− hγζk+1)>K (σk− hK ˜σk+1− hγζk+1) − hγ ˜σk+1> ζk+1 + h˜σ>k+1(wk+ ηkm) + h 2kw k+ ηmk k 2 ≤ −hσk>Kσk+ 2h2σ>kK (K ˜σk+1+ γζk+1) − hγ ˜σk+1> ζk+1+ h˜σ>k+1(wk+ ηkm) + h2kwk+ ηkmk 2 ≤ −hσk>Kσk− h γε − kwk+ ηmkk − 2hkKk 2 mkσkk k˜σk+1k + 2h2γkKkmkζk+1kkσkk +h 2 2 kwk+ η m k k 2, (78)
